Exponentially concave functions and a new information geometry

TL;DR

This paper introduces a new information geometry based on L-divergence derived from exponentially concave functions, linking optimal transport, dual geodesics, and applications in stochastic portfolio theory.

Contribution

It develops a novel information geometry on the simplex using L-divergence, revealing dual projective flatness and extending classical Pythagorean theorems with applications in finance.

Findings

L-divergence induces a new dualistic geometry on the simplex.

The geometry is dually projectively flat but not flat.

Displacement interpolation yields dual geodesics, with applications in portfolio optimization.

Abstract

A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two kinds of geodesics. We show that the induced geometry is dually projectively flat but not flat. Nevertheless, we prove an analogue of the celebrated generalized Pythagorean theorem from classical information geometry. On the other hand, we consider displacement interpolation under a Lagrangian integral action that is consistent with the optimal transport problem and show that the action minimizing curves are dual geodesics. The Pythagorean theorem is also shown to have an interesting application of determining the optimal trading frequency in stochastic portfolio theory.